Research Papers

J. Vib. Acoust. 2013;135(3):031001-031001-17. doi:10.1115/1.4023141.

Acoustic metamaterials are those structurally engineered materials that are composed of periodic cells designed in such a fashion to yield specific material properties (density and bulk modulus) that would affect the wave propagation pattern within in a specific way. All the currently exerted efforts are focused on studying passive metamaterials with fixed material properties. In this paper, the emphasis is placed on the development of a new class of composite one-dimensional active acoustic metamaterials (CAAMM) with effective densities and bulk moduli that are programmed to vary according to any prescribed patterns along its volume. A cylindrical water-filled cylinder coupled to two piezoelectric elements form a composite cell to act as a base unit for a periodic metamaterial structure. Two different configurations are considered. In the first configuration, a piezoelectric panel is flash-mounted to the face of the cylinder, while the other is side-mounted to the cylinder wall, introducing a variable stiffness along the wave propagation path. In the second configuration, the face-mounted piezoelectric panel remains unchanged, while the side-mounted panel is replaced with an active Helmholtz resonator with piezoelectric base panel. A detailed theoretical lumped-parameter model for the two configurations is present, from which the stiffness of both active elements is controlled via charge feedback control to yield arbitrary homogenized effective bulk modulus and density over a very wide frequency range. Numerical examples are presented to demonstrate the performance characteristics of the proposed. The CAAMM presents a viable approach to the development of effective domains with a controllable wave propagation pattern to suit many applications.

Commentary by Dr. Valentin Fuster
J. Vib. Acoust. 2013;135(3):031002-031002-14. doi:10.1115/1.4023139.

Experimental and modeling techniques for belt longitudinal static stiffness, longitudinal dynamic stiffness and damping coefficient, bending stiffness, and friction coefficient between a pulley and a belt are presented. Two methods for measuring longitudinal dynamic stiffness and damping coefficient of a belt are used, and the experimental results are compared. Experimental results show that the longitudinal dynamic stiffness of a belt is dependent on belt length, pretension, excitation amplitude and excitation frequency, and the damping coefficient of a belt is dependent on excitation frequency. Two models are presented to model the dependence of longitudinal dynamic stiffness and damping coefficient of a belt on belt length, pretension, excitation amplitude and excitation frequency. The proposed model is validated by comparing the estimated dynamic stiffness and damping with the experiment data. Also, the measurements of belt bending stiffness are carried out and the influences of the belt length on the belt bending stiffness are investigated. One test rig for measuring friction coefficient between a pulley and a belt are designed and fabricated, and the friction coefficient between the groove side belt with the groove side pulley, and the flat side belt with a flat pulley is measured with the test rig. The influences of wrap angle between pulley and belt, pretension of belt and rotational speed of the pulley on the friction coefficient are measured and analyzed. Taking an engine front end accessory drive system (FEAD) as the research example for the accessory drive system, experimental methods and the static and dynamic characteristics for the FEAD with seven pulleys, a tensioner, and a serpentine belt are presented.

Topics: Pulleys , Belts , Stiffness
Commentary by Dr. Valentin Fuster
J. Vib. Acoust. 2013;135(3):031003-031003-13. doi:10.1115/1.4023140.

This is the second part of the paper for modeling and validation of the rotational vibration responses for an accessory drive system. The unified formulas for modeling the rotational vibration of an accessory drive system are presented. In the modeling of an accessory drive system, the damping and stiffness of a belt are regarded as the function of the excitation frequency of an engine and the amplitude of belt stretching. Additionally, the creeping effect of a belt on the pulley wrap arc is included in the model. A general purpose software for calculating the rotational vibration of an accessory drive system is developed, based on the presented unified formulas. One accessory drive system with seven pulleys, a tensioner, and a serpentine belt is used as a studying example to demonstrate the unified formulas and the procedure for obtaining the rotational vibration. In the simulation of the accessory drive system, the stiffness and damping of the belt, the friction coefficient between the belt and pulley, and the excitation torques with multifrequency components from the crankshaft torsional vibration are obtained from the experiment in the first part of this paper. The static tension and steady-state tension of each belt span, along with the natural frequency of the accessory drive system, rotational vibrations of the driven pulley and tensioner arm, and the dynamic tension of the belt span are calculated and compared well with the experimental data, which validate the presented unified formulas and the developed general purpose software. The modeling method and the procedure described in this paper are instructive for designing an accessory drive system.

Commentary by Dr. Valentin Fuster
J. Vib. Acoust. 2013;135(3):031004-031004-12. doi:10.1115/1.4023300.

Planetary gears are widely used in the industry due to their advantages of compactness, high power-to-weight ratios, high efficiency, and so on. However, planetary gears such as that in wind turbine transmissions always operate under dynamic conditions with internal and external load fluctuations, which accelerate the occurrence of gear failures, such as tooth crack, pitting, spalling, wear, scoring, scuffing, etc. As one of these failure modes, gear tooth crack at the tooth root due to tooth bending fatigue or excessive load is investigated; how it influences the dynamic features of planetary gear system is studied. The applied tooth root crack model can simulate the propagation process of the crack along tooth width and crack depth. With this approach, the mesh stiffness of gear pairs in mesh is obtained and incorporated into a planetary gear dynamic model to investigate the effects of the tooth root crack on the planetary gear dynamic responses. Tooth root cracks on the sun gear and on the planet gear are considered, respectively, with different crack sizes and inclination angles. Finally, analysis regarding the influence of tooth root crack on the dynamic responses of the planetary gear system is performed in time and frequency domains, respectively. Moreover, the differences in the dynamic features of the planetary gear between the cases that tooth root crack on the sun gear and on the planet gear are found.

Commentary by Dr. Valentin Fuster
J. Vib. Acoust. 2013;135(3):031005-031005-14. doi:10.1115/1.4023143.

A numerical method using the multiple frequencies elliptical whirling orbit model and transient Reynolds-averaged Navier–Stokes (RANS) solution was proposed for prediction of the frequency dependent rotordynamic coefficients of annular gas seals. The excitation signal was the multiple frequencies waveform that acts as the whirling motion of the rotor center. The transient RANS solution combined with mesh deformation method was utilized to solve the leakage flow field in the annular gas seal and obtain the transient response forces on the rotor surface. Frequency dependent rotordynamic coefficients were determined by transforming the dynamic monitoring data of response forces and rotor motions to the frequency domain using the fast fourier transform. The frequency dependent rotordynamic coefficients of three types of annular gas seals, including a labyrinth seal, a fully partitioned pocket damper seal and a hole-pattern seal, were computed using the presented numerical method at thirteen or fourteen frequencies of 20–300 Hz. The obtained rotordynamic coefficients of three types of annular gas seals were all well agreement with the experimental data. The accuracy and availability of the proposed numerical method was demonstrated. The static pressure distributions and leakage flow rate of three types of annular gas seals were also illustrated.

Commentary by Dr. Valentin Fuster
J. Vib. Acoust. 2013;135(3):031006-031006-14. doi:10.1115/1.4023144.

This paper is the first work on the vibration of a high-speed rotating spherical shell that rotates about its symmetric axis by developing a set of motion governing equations with consideration of both the Coriolis and centrifugal accelerations as well as the hoop tension arising in the rotating shell due to the angular velocity. To the author's understanding, no such work has so far been published on the rotating spherical shell with the Coriolis and centrifugal accelerations as well as the hoop tension, although there have been the works published on the rotating hemispherical shell with consideration of the Coriolis and centrifugal forces. A thin rotating isotropic truncated circular spherical shell with the simply supported boundary conditions at both the ends is taken as an example for the free vibrational analysis. In order to validate the present formulation, comparisons are made with a nonrotating isotropic spherical shell, and a good agreement is achieved since no published data results from open literature are available for comparison on the dynamics of rotating spherical shell. By the Galerkin method, several case studies are conducted for investigation of the influence of the important parameters on the frequency characteristics of the rotating spherical shell. The parameters studied include the circumferential wave number, the rotational angular velocity, Young's modulus of the shell material, and the geometric ratio of the thickness to radius of the spherical shell.

Commentary by Dr. Valentin Fuster
J. Vib. Acoust. 2013;135(3):031007-031007-8. doi:10.1115/1.4023251.

In many small-scale devices, the materials employed are functionalized (doped) with microscale and/or nanoscale particles, in order to deliver desired overall dielectric properties. In this work, we develop a reduced-order lumped-mass model to characterize the dynamic response of a material possessing a microstructure that is comprised of an electromagnetically-neutral binder with embedded electromagnetically-sensitive (charged) particles. In certain industrial applications, such materials may encounter external electrical loading that can be highly oscillatory. Therefore, it is possible for the forcing frequencies to activate the inherent resonant frequencies of these micro- and nanostructures. In order to extract qualitative information, this paper first analytically investigates the mechanical and electromagnetic (cyclotronic) contributions to the dynamic response for a single particle, and then quantitatively investigates the response of a model problem consisting of a coupled multiparticle periodic array, via numerical simulation, using an implicit temporally-adaptive trapezoidal time-stepping scheme. For the model problem, numerical studies are conducted to observe the cyclotronically-dominated resonant frequency and associated beat phenomena, which arises due to the presence of mechanical and electromagnetic harmonics in the material system.

Commentary by Dr. Valentin Fuster
J. Vib. Acoust. 2013;135(3):031008-031008-11. doi:10.1115/1.4023145.

The vibrations of pipes and their supporting beams caused by the tank bulging mode are analyzed. A computationally efficient semianalytical approach is presented by introducing a local velocity potential for each pipe and developing a reduced order model for the frame structure consisting of the pipes and their supporting beams. To enable the analysis, the system of governing equations and boundary conditions is derived in a variational form. Numerical results show that large bending stress occurs in the supporting beams attached to the lower part of the tank wall. This bending stress can be reduced by decreasing the outer diameter of the supporting beams based on the fact that the dependence of the bulging-mode-induced displacements of the supporting beams on their stiffness is weak. This stress reduction method does not increase but decreases the stiffness of the supporting beams and therefore the cost, unlike many methods for improving structural reliability. A method for reducing the axial compressive stress in the supporting beams is also investigated.

Commentary by Dr. Valentin Fuster
J. Vib. Acoust. 2013;135(3):031009-031009-10. doi:10.1115/1.4023142.

A general method for predicting acoustic radiation from multiple periodic structures is presented and a numerical solution is proposed to find the radial displacement of thick laminated cylindrical shells with sparse cross stiffeners in the wavenumber domain. Although this method aims at the sound radiation from a single stiffened cylindrical shell, it can be easily adapted to analyze the vibrational and sound characteristics of two concentric cylindrical shells or two parallel plates with complicated periodic stiffeners, such as submarine and ship hulls. The sparse cross stiffeners are composed of two sets of parallel rings and one set of longitudinal stringers. The acoustic power of large cylindrical shells above the ring frequency is derived in the wavenumber domain on the basis of the fact that sound power is focused on the acoustic ellipse. It transpires that a great many band gaps of wave propagation in the helical wave spectra of the radial displacement for stiffened cylindrical shells are generated by the rings and stringers. The acoustic power and input power of stiffened antisymmetric laminated cylindrical shells are computed and compared. The acoustic energy conversion efficiency of the cylindrical shells is less than 10%. The axial and circumferential point forces can also produce distinct acoustic power. The radial displacement patterns of the antisymmetric cylindrical shell with fluid loadings are illustrated in the space domain. This study would help to better understand the main mechanism of acoustic radiation from stiffened laminated composite shells, which has not been adequately addressed in its companion paper (Cao et al., 2012, “Acoustic Radiation From Shear Deformable Stiffened Laminated Cylindrical Shells,” J. Sound Vib., 331(3), pp. 651-670).

Commentary by Dr. Valentin Fuster
J. Vib. Acoust. 2013;135(3):031010-031010-8. doi:10.1115/1.4023047.

A method of complex orthogonal decomposition is summarized for the time-domain, and then formulated and justified for application in the frequency-domain. The method is then applied to the extraction of modes from simulation data of sampled multimodal traveling waves for estimating wave parameters in one-dimensional continua. The decomposition is first performed on a transient nondispersive pulse. Complex wave modes are then extracted from a two-harmonic simulation of a dispersive medium. The wave frequencies and wave numbers are obtained by looking at the whirl of the complex modal coordinate, and the complex modal function, respectively, in the complex plane. From the frequencies and wave numbers, the wave speeds are then estimated, as well as the group velocity associated with the two waves. The decomposition is finally applied to a simulation of the traveling waves produced by a Gaussian initial displacement profile in an Euler–Bernoulli beam. While such a disturbance produces a continuous spectrum of wave components, the sampling conditions limit the range of modal components (i.e., mode shapes and modal coordinates) to be extracted. Within this working range, the wave numbers and frequencies are obtained from the extraction, and compared to theory. Modal signal energies are also quantified. The results are robust to random noise.

Commentary by Dr. Valentin Fuster
J. Vib. Acoust. 2013;135(3):031011-031011-7. doi:10.1115/1.4023816.

An investigation of noncontact manipulation techniques based on acoustic levitation was undertaken in air. The standing wave acoustic levitation (SWAL) was observed when standing waves trap small objects at pressure nodes. In this paper, two ultrasonic bolt-clamped Langevin type transducers (BLTs) generating traveling waves by modulating parameters of the two traveling waves were used to manipulate a trapped object. Frequency, amplitude, and phase modulations of the two actuators were exploited. From simulation and experiments, the phase modulation was prominent among other methods due to its long range and smooth operation. It is also found that angles between two actuators affect the trajectory of the trapped object during the parameter modulations. Sinusoidal and elliptic paths of the object were observed experimentally through a combination of parameters at certain tilt angles.

Commentary by Dr. Valentin Fuster
J. Vib. Acoust. 2013;135(3):031012-031012-18. doi:10.1115/1.4024087.

The effects of replacing rolling element bearings with journal bearings on the noise and vibration of a simple gearbox are computationally and experimentally evaluated. A modified component mode synthesis (CMS) approach is used, where the component modes of the shafting and gearbox housing are modeled using finite element analysis (FEA). Instead of using component modes with free boundary conditions, which is typical of CMS, the shafting and gearbox are coupled using nominal impedances computed for the different bearing types, improving convergence of the solution. Methods for computing the actual bearing impedances, including the high damping coefficients in journal bearings, are summarized. The sound radiated by the gearbox is computed using a boundary element (BE) model. The modeling results are validated against measurements made at the NASA Glenn Research Center. Both simulations and measurements reveal that the journal bearings, although highly damped, do not necessarily lead to strong reductions in gearbox vibration and noise.

Commentary by Dr. Valentin Fuster
J. Vib. Acoust. 2013;135(3):031013-031013-21. doi:10.1115/1.4023814.

Periodic impulses in vibration signals and its repeating frequency are the key indicators for diagnosing the local damage of rolling element bearings. A new method based on ensemble empirical mode decomposition (EEMD) and the Teager energy operator is proposed to extract the characteristic frequency of bearing fault. The signal is firstly decomposed into monocomponents by means of EEMD to satisfy the monocomponent requirement by the Teager energy operator. Then, the intrinsic mode function (IMF) of interest is selected according to its correlation with the original signal and its kurtosis. Next, the Teager energy operator is applied to the selected IMF to detect fault-induced impulses. Finally, Fourier transform is applied to the obtained Teager energy series to identify the repeating frequency of fault-induced periodic impulses and thereby to diagnose bearing faults. The principle of the method is illustrated by the analyses of simulated bearing vibration signals. Its effectiveness in extracting the characteristic frequency of bearing faults, and especially its performance in identifying the symptoms of weak and compound faults, are validated by the experimental signal analyses of both seeded fault experiments and a run-to-failure test. Comparison studies show its better performance than, or complements to, the traditional spectral analysis and the squared envelope spectral analysis methods.

Commentary by Dr. Valentin Fuster
J. Vib. Acoust. 2013;135(3):031014-031014-16. doi:10.1115/1.4023817.

A general wave approach for the vibration analysis of curved beam structures is presented. The analysis is based on wave propagation, transmission, and reflection, including the effects of both propagating and decaying near-field wave components. A matrix formulation is used that offers a systematic and concise method for tackling free and forced vibrations of complex curved beam structures. To illustrate the effectiveness of the approach, several numerical examples are presented. The predictions made using the wave approach are shown to be in excellent agreement with a conventional finite element analysis, with the advantage of reduced computational costs and good conditioning number of the characteristic equation. The developed wave approach is applied to investigate the free vibration, vibration transmission, and power flow of built-up structures consisting of curved beams, straight beams, and masses, with the aim for designing vibration isolation structure with high attenuation ability. Wave reflection and transmission in the infinite curved beam structure, as well as vibration and energy transmission in coupled finite curved beam structure are investigated. Numerical results show that wave mode conversion takes place for the reflected and transmitted wave propagating through a curved beam, and the power flow in the coupled curved beam structure shows energy attenuation and conversion by curved beam and the discontinuities. The investigation will shed some light on the designing of curved beam structures.

Commentary by Dr. Valentin Fuster
J. Vib. Acoust. 2013;135(3):031015-031015-9. doi:10.1115/1.4023812.

In this paper, acoustic wave propagation in a two- or three-dimensional phononic crystal consisting of Helmholtz resonators embedded in a fluid matrix is studied. The band structures are calculated to discuss the influence of the geometry topology of Helmholtz resonators on the bandgap characteristics. It is shown that a narrow bandgap will appear in the lower frequency range due to the resonance of the Helmholtz resonators. The width and position of this resonance bandgap can be tuned by adjusting the geometrical parameters of the Helmholtz resonator. The position of the resonance bandgap can be evaluated by the resonance frequency of the Helmholtz resonator. A decrease in the size of the opening generally results in a lower position and a smaller width of the bandgap. The system with one opening exhibits a wider bandgap in a lower position than the system with two openings.

Commentary by Dr. Valentin Fuster
J. Vib. Acoust. 2013;135(3):031016-031016-11. doi:10.1115/1.4023830.

The enhancement of Herschel–Quincke (HQ) waveguides to incorporate adaptive capabilities is investigated. Passive HQ waveguides are known to provide noise attenuation in pipes and ducts at selective narrow frequency bands associated with their resonances. The approach to achieve adaptation is to produce a frequency shift in these resonances to allow targeting incoming tonal noise of variable frequency. The frequency shift is obtained by placing a variable cross-section constriction along the interior of the waveguide. Two adaptive devices are considered. The first consists of a ball with fixed diameter that can be axially displaced inside the waveguide. Then, the frequency tuning is obtained as a function of the ball position. The second device consists of a diaphragm at fixed axial location which can be deformed to obtain a variable cross section. In this case, the frequency shift is obtained as a function of the diaphragm deflection. The internal acoustic dynamics of the two devices are investigated both analytically and experimentally. The created constriction inside the HQ waveguide is modeled as a series of constant cross-section tube elements with small finite area jump between adjacent pieces. The model is validated by comparing the predicted dynamics with experimental data from an extended impedance tube setup based on the two-microphone technique. Finally, attenuation predictions on a one-dimensional pipe are presented in order to illustrate the performance of the proposed adaptive HQ waveguides.

Commentary by Dr. Valentin Fuster
J. Vib. Acoust. 2013;135(3):031017-031017-11. doi:10.1115/1.4023843.

The stability and bifurcation of a flexible 3D rotor system are investigated in this paper. The rotor is discretized by 3D elements and reduced by using component mode synthesis. Periodic motions and stability margins are obtained by using the shooting method and path-following technique, and the local stability of the periodic motions is determined by using the Floquet theory. Comparisons indicate that 3D and 1D systems have a general resemblance in the bifurcation characteristics while mass eccentricity and rotating speed are changed. For both systems, the orbit size of the periodic motions has the same order of magnitude, and the vibration response has identical frequency components when typical bifurcations occur. The stress distribution and location of the maximum stress spot are determined by the bending mode of the rotor. The type of 3D element has a slight effect on the stability and bifurcation of the rotor system. Generally, this paper presents a feasible method for analyzing the stability and bifurcation of complex rotors without much structural simplification.

Commentary by Dr. Valentin Fuster

Technical Briefs

J. Vib. Acoust. 2013;135(3):034501-034501-5. doi:10.1115/1.4023052.

Combi-bearing is a combined thrust-journal bearing design used in vertical hydropower rotors. The dynamic characteristics of this component (combi-bearing) were analytically modeled by Luneno et al. (2011, “Model Based Analysis of Coupled Vibrations Due to the Combi-Bearing in Vertical Hydroturbogenerator Rotors,” ASME J. Vib. Acoust., 133, p. 061012). This analytic model was inserted into a finite element model of a vertical rotor rig and numerically simulated. In this paper, the simulated vertical rotor-bearings system is a small-scale vertical machine constructed to validate the analytically derived combi-bearing model. Good agreement was found between the simulation and experimental results. The simulation and experimental results showed that the journal (radial) bearing's position relative to the contact point between the combi-bearing's collar and the rotor influences the rotor system's fundamental natural frequencies. Therefore, the combi-bearing model needs to be included into rotor dynamic models. Neglecting the effect of this component may cause significant errors in the predicted results.

Topics: Bearings , Rotors
Commentary by Dr. Valentin Fuster
J. Vib. Acoust. 2013;135(3):034502-034502-4. doi:10.1115/1.4023146.

An analytical method is derived for the vibration analysis of doubly curved shallow shells with arbitrary elastic supports alone its edges, a class of problems which are rarely attempted in the literature. Under this framework, all the classical homogeneous boundary conditions for both in-plane and out-of-plane displacements can be universally treated as the special cases when the stiffness for each of restraining springs is equal to either zero or infinity. Regardless of the boundary conditions, the displacement functions are invariably expanded as an improved trigonometric series which converges uniformly and polynomially over the entire solution domain. All the unknown expansion coefficients are treated as the generalized coordinates and solved using the Rayleigh–Ritz technique. Unlike most of the existing solution techniques, the current method offers a unified solution to a wide spectrum of shell problems involving, such as different boundary conditions, varying material and geometric properties with no need of modifying or adapting the solution schemes and implementing procedures. A numerical example is presented to demonstrate the accuracy and reliability of the current method.

Commentary by Dr. Valentin Fuster

Technical Brief

J. Vib. Acoust. 2013;135(3):034503-034503-5. doi:10.1115/1.4023834.

In this note, a closed-form solution of periodic motions in a periodically forced oscillator with quadratic nonlinearity is presented without any small parameters. The perturbation method is based on one harmonic term plus perturbation modification, and the traditional harmonic balance is to arbitrarily select harmonic terms with constant coefficients. If harmonic terms are not enough included in the approximate solution, such a solution is not an appropriate, analytical solution for periodic motions, and some analytical solutions cannot be caught.

Commentary by Dr. Valentin Fuster

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